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Exam2_Solutions_W11 - Math 115 Second Midterm Name EXAM...

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Math 115 — Second Midterm March 22, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 8 pages including this cover. There are 7 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 ′′ × 5 ′′ note card. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. 9. Note that problems 5–7 will be graded giving very little partial credit. Problem Points Score 1 15 2 16 3 12 4 15 5 15 6 15 7 12 Total 100
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Math 115 / Exam 2 (March 22, 2011) page 2 w 1 . [15 points] A hoophouse is an unheated greenhouse used to grow certain types of vegetables during the harsh Michigan winter. A typical hoophouse has a semi-cylindrical roof with a semi-circular wall on each end (see figure to the right). The growing area of the hoophouse is the rectangle of length and width w (each measured in feet) which is covered by the hoophouse. The cost of the semi-circular walls is $0.50 per square foot and the cost of the roof, which varies with the side length , is $1 + 0 . 001 per square foot. a . [4 points] Write an equation for the cost of a hoophouse in terms of and w . (Hint: The surface area of a cylinder of height and radius r , not including the circles on each end, is A = 2 πrℓ .) Solution: The roof has area πrℓ = π 2 wℓ . The walls have area πr 2 = π 4 w 2 . This means the cost is C = 0 . 50 · π 4 w 2 + (1 + 0 . 001 ) π 2 wℓ = π 8 w 2 + π 2 (1 + 0 . 001 ) wℓ. b . [11 points] Find the dimensions of the least expensive hoophouse with 8000 square feet of growing area.
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